1. Firstly you should have a Supplier questionnaire to get to know your supplier. In the questionnaire you should ask for information if they have full traceability, if they have implemented HACCP based management system. whether they have any certificates etc.
2. You should have Technical Data Sheets for all raw material you are using
3. You should have Material Safety Data Sheets for all chemical raw materials (ink. adhesive, finihing agents, solvents etc.)
4. In Europe we have "Declaration of Conformity" according to EU regulations. They also include information about FDA in most cases. You should have this document as well. (Or may be there is something comparable in FDA requirements I am not familiar with that)
In some cases you can make exceptions, for example if the raw material is not on a food contact surface you can skip number 4.
Once you have these in hand you have to decide your approval criteria. For example if you are making food packaging you must have number 4.
Or if you are dealing with very big suppliers they won't bother filling out your supplier questionnaire, so you should be able to make an exception if you have the documents proving the safety of the raw material.
You might want to implement a system where you always get a trial order and approve it based on your trial. It is up to you.
You can also give the suppliers points based on delivery time, service, price etc. and have a scoring system.
These are just ideas. I am sure you can find some examples of it in the forum.